# Vertex-Constraints in 3D Higher Spin Theories

## TL;DR

This paper investigates gauge invariance constraints in 3D higher-spin gravities, revealing that higher-order interactions are limited to scalars and Maxwell fields, with full non-linear interactions determined by cubic vertices.

## Contribution

It demonstrates that in 3D higher-spin theories, higher-order vertices independent of cubic ones are restricted to scalars and Maxwell fields, fixing the non-linear interactions.

## Key findings

- Higher-order vertices depend only on scalars and Maxwell fields.
- Full non-linear interactions are determined by the cubic vertex.
- Constraints from gauge invariance limit possible interactions.

## Abstract

We analyse the constraints imposed by gauge invariance on higher-order interactions between massless bosonic fields in three-dimensional higher-spin gravities. We show that vertices of quartic and higher order that are independent of the cubic ones can only involve scalars and Maxwell fields. As a consequence, the full non-linear interactions of massless higher-spin fields are completely fixed by the cubic vertex.